Sequences, Series, and Convergence

First, a list; then, a sum

Two words sound almost the same but mean very different things, and keeping them apart is half the battle. A sequence is simply an ordered, never-ending list of numbers: a first term, a second, a third, and so on forever. For example 1, 1/2, 1/3, 1/4, ... is a sequence — nobody is adding anything yet, we are just listing the values one after another.

An infinite series is what you get when you decide to add all the terms of a sequence together: 1 + 1/2 + 1/3 + 1/4 + ... A sequence is a list; a series is the running total of that list. The deep question — the one this whole guide is about — is whether such an endless addition can ever settle on a real, finite number.

What 'converge' really means: partial sums

You cannot literally perform infinitely many additions. So mathematicians do something cleverer: they add just the first term, then the first two, then the first three, and watch where these partial sums are heading. The n-th partial sum S_n = a_1 + a_2 + ... + a_n is an honest, finite addition you can actually do. Now you have a brand-new sequence — S_1, S_2, S_3, ... — and you can ask the question you already know how to ask about a sequence.

Here is the precise definition, and it leans entirely on the idea of a limit you met earlier. A series converges if its sequence of partial sums approaches a single finite number L as n grows without bound; that number L is then defined to be the sum of the series. If the partial sums never settle — they march off to infinity, or oscillate forever — the series diverges and simply has no sum. This is the heart of convergence and divergence.

series:        a_1 + a_2 + a_3 + a_4 + ...
partial sums:  S_1 = a_1
               S_2 = a_1 + a_2
               S_3 = a_1 + a_2 + a_3
               ...
converges  <=>  lim (n -> infinity) S_n = L   (a finite number)

Two famous series: one tidy, one treacherous

The friendliest series is the geometric series, where each term is the previous one multiplied by a fixed ratio r: a + a*r + a*r^2 + a*r^3 + ... Think of repeatedly taking half of what is left. Astonishingly, when the ratio is small enough — when |r| < 1 — the infinite total is exactly a / (1 - r), a clean finite formula. For 1/2 + 1/4 + 1/8 + ... we have a = 1/2 and r = 1/2, so the sum is (1/2) / (1 - 1/2) = 1. The whole endless pile adds up to a humble 1.

geometric series:   a + a*r + a*r^2 + a*r^3 + ...

   if |r| < 1:   sum = a / (1 - r)     (converges)
   if |r| >= 1:  no finite sum         (diverges)

example:   1/2 + 1/4 + 1/8 + 1/16 + ...  =  (1/2)/(1 - 1/2)  =  1

Now the treacherous one. The harmonic series 1 + 1/2 + 1/3 + 1/4 + ... has terms that shrink to zero, so surely its sum is finite? No — it diverges to infinity. The terms simply do not shrink fast enough; the partial sums creep upward without bound, just very slowly. This is the famous trap: terms going to zero is necessary for convergence but nowhere near sufficient.

Zeno's runner, finally caught

Twenty-five centuries ago, Zeno argued that motion is impossible. To reach the finish line a runner must first cover half the distance, then half of what remains, then half of that, forever — infinitely many steps. Surely infinitely many steps take infinitely long? The puzzle stumped philosophers for ages because they lacked exactly the idea you now have: a convergent series.

1. Write the distances as a geometric series: 1/2 + 1/4 + 1/8 + 1/16 + ... — each step covers half of what was left.
2. Look at the partial sums: 1/2, then 3/4, then 7/8, then 15/16, ... They creep ever closer to 1 but never overshoot it.
3. Take the limit: since |r| = 1/2 < 1, the sum is a / (1 - r) = (1/2) / (1 - 1/2) = 1. The infinitely many steps add up to exactly one whole distance.
4. The same logic applies to the time taken: infinitely many ever-shorter durations also sum to a finite time. So the runner crosses the line right on schedule.